I am mining blocks that don't drop as the block itself (ex. stone drops cobblestone, redstone ore drops redstone dust, etc.), but that doesn't happen. When I mine stone it drops stone instead of cobblestone and when I mine redstone ore it drops the ore block and not redstone dust.
Your pickaxe probably has silk touch on it.
The Silk Touch enchantment allows you to mine blocks themselves instead of getting the usual dropped items. With this enchantment, you can gather items that are fragile and break such as glass or ice. Or you can mine and gather the block itself such as diamond ore or coal ore (instead of getting diamond and coal drops).
You can add the Silk Touch enchantment to any pickaxe, shovel, or axe using an enchantment table, anvil, or game command. Then use the enchanted tool to mine and see just how quickly you can add those rare and fragile items to your inventory!!
The maximum level for the Silk Touch enchantment is Level 1. This means that you can only enchant an item with up to Silk Touch I, and nothing higher for this enchantment.
This issue is most likely because you have the "Silk Touch" enchantment. According to the wiki, silk touch is an enchant that 'Allows collection of blocks that are normally unobtainable: Mined blocks drop themselves instead of the usual items." Unfortunately, this cannot be undone, as the enchantment cannot be removed from the pickaxe. The best way to solve this is by getting another pickaxe.